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The value of cot (?_n=1²³ cot^-1 (1 + ?_k=1ⁿ 2k)) is

1) 23 / 25 2) 25 / 23 3) 23 / 24 4) 24 / 23 Solution: (2) 25 / 23 ∑k=1n 2k = 2 [1 + 2 + 3 + …. + n] = 2 [(n (n + 1)] / 2 = (n2 +... View Article

The principal value of sin^-1 {tan (- 5? / 4)} is

1) π / 4 2) - (π / 4) 3) π / 2 4) - (π / 2) Solution: (4) - (π / 2) sin-1 {tan (- 5π / 4)} = sin-1{- tan (π + π / 4)} = sin-1 (-... View Article

If (tan^-1 x)² + (cot^-1 x)² = 5?² / 8, then x is equal to

1) 0 2) 2 3) 1 4) - 1 5) 2√2 Solution: (4) - 1 (tan-1 x)2 + (cot-1 x)2 = 5π2 / 8 (tan-1 x)2 + [(π / 2) - tan-1 x]2 = 5π2 / 8... View Article

If 5 cos^-1 [1 – x²] / [1 + x²] + 7 sin^-1 2x / [1 + x²] – 4 tan^-1 2x / [1 – x²] – tan^-1 x = 5?, then x =

1) 3 2) -√3 3) √2 4) 2 5) √3 Solution: (5) √3 5 cos-1 [1 - x2] / [1 + x2] + 7 sin-1 2x / [1 + x2] - 4 tan-1 2x / [1 - x2] -... View Article

sin (2 sin^-1 ?63 / 65) =

1) 2√126 / 65 2) 4√65 / 65 3) 8√63 / 65 4) √63 / 65 Solution: (1) 2√126 / 65 sin (2 sin-1 √63 / 65) = sin (sin-1 2 * √63 / 65 * √1... View Article

If tan^-1 (1 / 3) + tan^-1 (3/4) – tan^-1 (x / 3) = 0, then x is equal

1) 7/3 2) 3 3) 11/3 4) 13/3 Solution: (4) 13 / 3 tan-1 (1 / 3) + tan-1 (3/4) - tan-1 (x / 3) = 0 tan-1 (1 / 3) + tan-1 (3 / 4) =... View Article

The solution of tan^-1 x + 2 cot^-1 x = 2? / 3 is

1) - (1/√3) 2) 1/√3 3) -√3 4) √3 Solution: (4) √3 tan-1 x + 2 cot-1 x = 2π / 3 tan-1 x + 2 [(π / 2) - tan-1 x] = 2π / 3 - tan-1... View Article

If ? = sin^-1 (?3 / 2) + sin^-1 (1 / 3) and ? = cos^-1 (?3 / 2) + cos^-1 (1 / 3), then

1) α > β 2) α = β 3) α < β 4) α + β = 2π Solution: (3) α < β α = sin-1 (√3 / 2) + sin-1 (1 / 3) and β = cos-1 (√3 / 2) + cos-1 (1 /... View Article

The simplified expression of sin (tan^-1 x), for any real number x is given by

1) 1 / √1 + x2 2) x / √1 + x2 3) - 1 / √1 + x2 4) - x / √1 + x2 Solution: (2) x / √1 + x2 Let tan-1 x = θ x = tan θ = sin θ / cos... View Article

If sin^-1 (3 / x) + sin^-1 (4 / x) = ? / 2, then x =

1) 3 2) 5 3) 7 4) 11 Solution: (2) 5 sin-1 (3 / x) + sin-1 (4 / x) = π / 2 sin-1 (3x) = cos-1 (4x) sin-1 (3x) = sin-1 (√x2 - 16 / x) 3 / x =... View Article

The value of cot (cosec^-1 (5 / 3) + tan^-1 (2 / 3)) is

1) 5/17 2) 6/17 3) 3/17 4) 4/17 Solution: (2) 6 / 17 Let (cosec-1 (5 / 3) = θ, tan-1 (2 / 3) = ɸ cosec θ = 5 / 3 and tan ɸ = 2 /... View Article

If 0 < x < 1, then ?1 + x² [{x cos (cot^-1 x)} + sin (cot^-1 x)}² – 1]^½ =

1) x / √1 + x2 2) x 3) x √1 + x2 4) √1 + x2 Solution: (3) x √1 + x2 0 < x < 1 (cot-1 x) = θ cot θ = x sin θ = 1 / √1 + x2... View Article

The derivative of F[f{?(x)}] is

(1) F’[f{φ(x)}] (2) F’[f{φ(x)}] f’{φ(x)} (3) F’[f{φ(x)}] f’{φ(x)} φ’(x) (4) none of these Solution: Let y = F[f{φ(x)}]... View Article

The differential coefficient of f(log x) when f(x) = log x is

(1) x log x (2) x/log x (3) 1/x log x (4) log x/x Solution: Given f(x) = log x f(log x ) = log log x Differentiate w.r.t.x... View Article

Let f and g be differentiable function satisfying g’(a) = 2, g(a) = b and fog = I(Identity function). Then f’(b) is equal to

(1) 1/2 (2) 2/3 (3) 2 (4) none of these Solution: Given fog = I(identity function) So f(g(x)) = x Differentiate w.r.t.x... View Article

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